On compositions of the loop and suspension functors

Abstract:

The problem studied is whether, from knowledge of the homotopy type of ${\Omega ^{{d_k}}}{\Sigma ^{{c_k}}} \cdots {\Omega ^{{d_2}}}{\Sigma ^{{c_2}}}{\Omega ^{{d_1}}}{\Sigma ^{{c_1}}}X = MX$ for suitable spaces X, one can recover the nonnegative integers ${c_1},{d_1}, \ldots ,{c_k},{d_k}$. The Betti numbers of X and ${c_1},{d_1}, \ldots ,{c_k},{d_k}$ do determine the ith Betti number of MX, but even for small k , i and for X a sphere (say) the answer is a complicated one, since it depends on parities and graded Witt numbers depending on graded Witt numbers. It is shown that k can be found and that ${c_i},{d_j}$ can always be determined up to finitely many possibilities and usually uniquely.